Precalculus Examples

$f (x) = - 3 x^{2} + 15 x + 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- 3 x^{2} + 15 x + 1$ .

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Step 1.1.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 3$

$b = 15$

$c = 1$

Step 1.1.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{15}{2 \cdot - 3}$

Step 1.1.3.2

Simplify the right side.

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Step 1.1.3.2.1

Cancel the common factor of $15$ and $- 3$ .

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Step 1.1.3.2.1.1

Factor $3$ out of $15$ .

$d = \frac{3 (5)}{2 \cdot - 3}$

Step 1.1.3.2.1.2

Cancel the common factors.

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Step 1.1.3.2.1.2.1

Factor $3$ out of $2 \cdot - 3$ .

$d = \frac{3 (5)}{3 (2 \cdot - 1)}$

Step 1.1.3.2.1.2.2

Cancel the common factor.

$d = \frac{3 \cdot 5}{3 (2 \cdot - 1)}$

Step 1.1.3.2.1.2.3

Rewrite the expression.

$d = \frac{5}{2 \cdot - 1}$

Step 1.1.3.2.2

Multiply $2$ by $- 1$ .

$d = \frac{5}{- 2}$

Step 1.1.3.2.3

Move the negative in front of the fraction.

$d = - \frac{5}{2}$

Step 1.1.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 1 - \frac{15^{2}}{4 \cdot - 3}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise $15$ to the power of $2$ .

$e = 1 - \frac{225}{4 \cdot - 3}$

Step 1.1.4.2.1.2

Multiply $4$ by $- 3$ .

$e = 1 - \frac{225}{- 12}$

Step 1.1.4.2.1.3

Cancel the common factor of $225$ and $- 12$ .

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Step 1.1.4.2.1.3.1

Factor $3$ out of $225$ .

$e = 1 - \frac{3 (75)}{- 12}$

Step 1.1.4.2.1.3.2

Cancel the common factors.

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Step 1.1.4.2.1.3.2.1

Factor $3$ out of $- 12$ .

$e = 1 - \frac{3 \cdot 75}{3 \cdot - 4}$

Step 1.1.4.2.1.3.2.2

Cancel the common factor.

$e = 1 - \frac{3 \cdot 75}{3 \cdot - 4}$

Step 1.1.4.2.1.3.2.3

Rewrite the expression.

$e = 1 - \frac{75}{- 4}$

Step 1.1.4.2.1.4

Move the negative in front of the fraction.

$e = 1 - - \frac{75}{4}$

Step 1.1.4.2.1.5

Multiply $- - \frac{75}{4}$ .

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Step 1.1.4.2.1.5.1

Multiply $- 1$ by $- 1$ .

$e = 1 + 1 (\frac{75}{4})$

Step 1.1.4.2.1.5.2

Multiply $\frac{75}{4}$ by $1$ .

$e = 1 + \frac{75}{4}$

Step 1.1.4.2.2

Write $1$ as a fraction with a common denominator.

$e = \frac{4}{4} + \frac{75}{4}$

Step 1.1.4.2.3

Combine the numerators over the common denominator.

$e = \frac{4 + 75}{4}$

Step 1.1.4.2.4

Add $4$ and $75$ .

$e = \frac{79}{4}$

Step 1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x - \frac{5}{2})}^{2} + \frac{79}{4}$ .

$- 3 {(x - \frac{5}{2})}^{2} + \frac{79}{4}$

Step 1.2

Set $y$ equal to the new right side.

$y = - 3 {(x - \frac{5}{2})}^{2} + \frac{79}{4}$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 3$

$h = \frac{5}{2}$

$k = \frac{79}{4}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{5}{2}, \frac{79}{4})$

Step 4